Properties of Kloosterman sums for long Weyl element on GL(n)

We give an alternate expression for the Kloosterman sums associated with the long Weyl element of $GL(n)$ that avoides the use of Plücker coordinates. The expression is instead based on the use of the well-known Cauchy-Binet formula from linear algebra. From this expression, we prove some important properties of these Kloosterman sums.